Order Ideals and Filters

Let $P$ be a poset. A subset $I$ of $P$ is said to be an order ideal if

• $I$ is a lower set: $\down I=I$ and
• $I$ is a directed set: $I$ is non-empty, and every pair of elements in $I$ has an upper bound in $I$

Given a subset $A$ of a poset $P$ we say that $B$ is the ideal generated by $A$ if $B$ is the smallest order ideal (of $P$ containing $A$ $B$ is denoted by $\langle A\rangle$ Note that $\langle A\rangle$ exists iff $A$ is a directed set. In particular, for any $x\in P$ $\down x$ is the ideal generated by $x$ Also, if $P$ is an upper semilattice, then for any $A\subseteq P$ let $A'$ be the set of finite joins of elements of $A$ then $A'$ is a directed set, and $\langle A\rangle=\down A'$

Dually, an order filter (or simply a filter) in $P$ is a non-empty subset $F$ which is both an upper set and a filtered set (every pair of elements in $F$ has a lower bound in $F$ . A principal filter is a filter of the form $\up x$ for some $x\in P$

Remark. This is a generalization of the notion of a filter in a set. In fact, both ideals and filters are generalizations of ideals and filters in semilattices and lattices.

Examples in a Semilattice

A subset $I$ in an upper semilattice $P$ is a semilattice ideal if

1. if $a,b\in I$ then $a\vee b\in I$ (condition for being an upper subsemilattice)
2. if $a\in I$ and $b\le a$ then $b\in I$

Then the two definitions are equivalent: if $P$ is an upper semilattice, then $I\subseteq P$ is a semilattice ideal iff $I$ is an order ideal of $P$ if $I$ is a semilattice ideal, then $I$ is clearly a lower and directed (since $a\vee b$ is an upper bound of $a$ and $b$ ; if $I$ is an order ideal, then condition 2 of a semilattice ideal is satisfied. If $a,b\in I$ then there is a $c\in I$ that is an upper bound of $a$ and $b$ Since $I$ is lower, and $a\vee b\le c$ we have $a\vee b\in I$

Going one step further, we see that if $P$ is a lattice, then a lattice ideal is exactly an order ideal: if $I$ is a lattice ideal, then it is clearly an upper subsemilattice, and if $b\le a\in I$ then $b=a\wedge b\in I$ also, so that $I$ is a semilattice ideal. On the other hand, if $I$ is a semilattice ideal, then $I$ is an upper subsemilattice, as well as a lower subsemilattice, for if $a\in I$ then $a\wedge b\in I$ as well since $a\wedge b\le a$ This shows that $I$ is a lattice ideal.

Dually, we can define a filter in a lower semilattice, which is equivalent to an order filter of the underly poset. Going one step futher, we also see that a lattice filter in a lattice is an order filter of the underlying poset.

Remark. An alternative but equivalent characterization of a semilattice ideal $I$ in an upper semilattice $P$ is the following: $a,b\in I$ iff $a\vee b\in I$